I recently sent a question to the BATF asking if there was a national database
of stolen firearms.
This is what I received back:

Thank you for your inquiry to the Bureau of Alcohol, Tobacco, Firearms
and Explosives (ATF). ATF does not maintain a national database of
stolen or lost firearms. Generally, when receiving a firearms
theft/loss report, your State and local police agencies will record the
stolen/lost firearm information with the National Crime Information
Center (NCIC). The NCIC is maintained by the Federal Bureau of
Investigations (FBI). Should you wish to conduct an inquiry of a
particular firearm to determine if it has been lost or stolen, your
State and local police agencies may be able to assist you with this
matter.

It amazes me that the BATF does not seem to think that it is important to
maintain a list of lost or stolen firearms.
Firearms can be sold out of state,
which means it can be difficult to determine if a firearm is lost or stolen.
It is not reasonable to expect a person to call all 50 States to ensure that a
firearm has not been stolen.

I would think that it would be common sense for the BATF to maintain a list of
stolen firearms so that it is easier to track the firearms and convict
criminals,
as well as ensuring that firearms dealers are not dealing in stolen firearms.

As insane as it seems,
it looks like the BATF is making is easier for criminals to sell stolen
firearms by not making it easy for people to ensure a firearm is not stolen.

Update:

It turns out that the BATF does have a database of stolen firearms,
but it is only available to Law Enforcement.